We report measurements of transverse momentum $p_t$ spectra for ten event multiplicity classes of p-p collisions at $\sqrt{s} = 200$ GeV. By analyzing the multiplicity dependence we find that the spectrum shape can be decomposed into a part with amplitude proportional to multiplicity and described by a L\'evy distribution on transverse mass $m_t$, and a part with amplitude proportional to multiplicity squared and described by a gaussian distribution on transverse rapidity $y_t$. The functional forms of the two parts are nearly independent of event multiplicity. The two parts can be identified with the soft and hard components of a two-component model of p-p collisions. This analysis then provides the first isolation of the hard component of the $p_t$ spectrum as a distribution of simple form on $y_t$.
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